A380322 a(n) is the sum of exponentially odd divisors of n^2.

1, 3, 4, 11, 6, 12, 8, 43, 31, 18, 12, 44, 14, 24, 24, 171, 18, 93, 20, 66, 32, 36, 24, 172, 131, 42, 274, 88, 30, 72, 32, 683, 48, 54, 48, 341, 38, 60, 56, 258, 42, 96, 44, 132, 186, 72, 48, 684, 351, 393, 72, 154, 54, 822, 72, 344, 80, 90, 60, 264, 62, 96, 248

Offset: 1

Author

Amiram Eldar, Jan 20 2025

Keywords

• nonn
• easy
• mult

Comments

The number of exponentially odd divisors of n^2 is equal to the number of divisors of n, A000005(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A000290, A005117, A033634, A268335.

Cf. A202535.

Programs

Mathematica

f[p_, e_] := (p^(2*e+1) - p)/(p^2 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e+1) - p)/(p^2 - 1) + 1);}

Formula

a(n) = A033634(A000290(n)) = A033634(n^2).

a(n) >= A000203(n), with equality if and only if n is squarefree (A005117).

Multiplicative with a(p^e) = (p^(2*e+1) - p)/(p^2 - 1) + 1.

Dirichlet g.f.: zeta(s-2) * zeta(s) * Product_{p prime} (1 - 1/p^(s-2) + 1/p^(s-1)).